Neurophilosophy of Number

ABSTRACT

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism.

Acknowledgements

Since my article, ‘Philosophie de la neuropsychologie du nombre’, published in 2014 in Intellectica and for which I benefited from the remarks of John Stewart, I have had the opportunity to expound roughly the same content in the workshop, ‘Philosophy of Mathematics and Logic’, organised by José Ferreirós at the Universidad de Sevilla, the proceedings of which were published in 2016. Different versions constituted the content of a talk in the ANR-DFG workshop, ‘Mathematics: Objectivity by Representation’, organised by Hanns Leitgeb and Gerhard Heinzmann at the Ludwig-Maximilians-Universität München (November 2014), and of another talk in the Colloquium de Mathématiques directed by Henri Lombardi at the Université de Franche-Comté, Besançon (March 2015). I still talked about the same subject at the thematic trimester ‘Current Issues in the Philosophy of Practice of Mathematics and Informatics' organised by Sébastien Maronne at the Mathematical Institute of the Université de Toulouse. Questions and comments from the audience in general and from Jeremy Avigad and Paolo Mancosu in particular led me to deepen my philosophical analysis and to present a completely new version at the 43rd Annual Philosophy of Science Conference, Inter-University Centre, Dubrovnik, April 2016. I thank Michel Ghins for inviting me to this conference, the participants for their useful comments, and James W. McAllister for inviting me to turn the talk into a paper for International Studies in the Philosophy of Science.

Notes

1 ‘Cognitive’ means pertaining to the action or process of knowing and is used for ‘any kind of mental operation or structure that can be studied in precise terms’ (Lakoff and Johnson 1999, 11). So it should not be confused with the narrow concept used in some traditions of analytic philosophy, where ‘cognitive’ refers only to formal rules and truth conditional semantics.

2 Nativism is sometimes contested, for instance in Stewart (1993), Elman et al. (1996), Palmer (2000), and Lécuyer and Durand (2012).

3 In particular, contrary to Piaget’s theory, ‘out of sight’ is not ‘out of mind’.

4 Renée Baillargeon and Elizabeth Spelke introduced the ‘violation-of-expectation looking-time methodology’ in the mid-1980s. Samples of experiments using this methodology are given in Carey (2009, 40–48).

5 It has been alternatively suggested that the approximate number system supports also the understanding of ordinal relation (for instance McCrink and Birdsall 2015, 263).

6 Alternatively, Barsalou (1999, 2008) and Campbell (2015) argue that elementary arithmetic is not based on amodal representations and they emphasize embodied cognitive processes.

7 Dehaene (2011, 49) mentions another experiment: watching a picture with two forks, one fork broken into two pieces, a 3- or 4-year-old child counts the broken fork twice and says that the total number of forks equals three.

8 For a discussion of technical aspects see Harvey et al. (2013) and Campbell (2015).

9 However some experiments show that the SNARC is linked with the ordinal position in a sequence rather than with the cardinal aspect of things. An alternative to Dehaene’s interpretation is given in van Dijck et al. (2015).

10 Brian C. Goodwin defended a non-reductionist structural view and criticized the excesses of Neo-Darwinism: see in particular Goodwin (1993, 1994).

11 Spelke (1983, 1985, 1990, 1993): ‘Infants divide perceptual arrays into units that move as connected wholes, that move separately from one another, that tend to maintain their size and shape over motion, and that tend to act upon each other only on contact.’

12 Dehaene adopted the two-systems view: Feigenson, Dehaene, and Spelke (2004), Revkin et al. (2008).

13 The model of object-files was proposed in Treisman, Kahneman, and Burkell (1983), Kahnemann, Treisman, and Gibbs (1992), Pylyshyn and Storm (1998), and Pylyshyn (2003a).

14 This distinction appeared first in Pylyshyn (2003b).

15 ‘Focal attention is typically directed to objects rather than to places and therefore the earliest stages of vision are concerned with individuating objects and that when visual properties are encoded they are encoded as properties of individual objects’ (Pylyshyn 2003a, 4, 13; emphasis in the original).

16 By contrast, according to Pylyshyn object tracking is engraved in the visual architecture; therefore the spatio-temporal parameters of the object do not come into play.

17 It has been argued that set representations are required for acquisition of the number concept: Halberda and Feigenson (2008). Thus the set-theoretical model is supposed to mirror the real development of children; though this model was invented in the nineteenth century by Bernard Bolzano, Richard Dedekind, and Georg Cantor, it was taught very recently. As Hodes (2008) notes, cognitive scientists are ‘overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians)'.

18 See also Burge (2011), 125: ‘There is substantial evidence that perceptual body representations occur in the visual systems of many mammals and some birds. Anticipations of continuities that are relevant to perceiving entities as bodies are associated with very early vision. The anticipations are not matters of conception or prediction’.

19 See also Barsalou (1999, 2008) for support of the view that categorization is grounded in the sensorimotor regions of the brain, and Lakoff and Johnson (1999) for an argument that abstract concepts are grounded metaphorically in embodied and situated knowledge.

20 For alternative views see e.g. Weiskopf (2008): percept and concept are not ontologically but functionally distinct, and Weiskopf (2009): concepts are not a single, uniform kind of psychological entity, but are constituted by multiple representational kinds, with the particular kind of concept used on an occasion being determined by properties of the context.

21 According to Barsalou’s ‘grounded cognition’, cognition does not reside in a separate semantic memory system; it shares mechanisms with the brain’s modal systems for perception, action and introspection (Barsalou 2008). By contrast, Núñez argues that numbers are not hard-wired. According to him, the leap to number concepts proper relies, in part, on two embodied, domain-general cognitive mechanisms: conceptual metaphor and fictive motion. Conceptual metaphor is both a particular inference-preserving cross-domain mapping, and the cognitive mechanism that enables such mapping. Fictive motion is a cognitive mechanism through which we unconsciously and effortlessly conceptualize static entities in dynamic terms (Lakoff and Núñez 2000; Núñez 2009; Núñez and Marghetis 2015). See also Harnad (1987), Neisser (1987), Medin (1989); Rips, Bloomfield, and Asmuth (2008).

22 Actually, a big amount of logical and metaphysical contemporary reflections is devoted to examining the relations between explanation and grounding. See for instance Correia and Schnieder (2012). Taking into account such current research would help to make philosophical interpretations of scientific experiments more cautious.